Finite impulse response (FIR) filters are commonly used digital filters.
An FIR filter has an impulse response that settles to zero in a finite number of sample periods. FIR filters are inherently stable because FIR filters require no feedback and have their poles at the origin (within the unit circle of the complex z plane). However, all digital filters, including FIR filters, are sensitive to perturbations in the filter's tap coefficients.
A digital filter constructed as a cascade of two or more subfilters can possess the capability of lowering the filter's sensitivity to these filter coefficient perturbations. This property is described in J. W. Adams and A. N. Willson, Jr., “A new approach to FIR digital filters with fewer multipliers and reduced sensitivity,” IEEE Trans. Circuits Syst., vol. CAS-30, pp. 277-283, May 1983 [referred to herein as “Adams”] which is herein incorporated by reference in its entirety.
In general, during the design of an FIR filter, the filter's length and tap coefficient's are selected to meet pre-defined characteristics that are usually specified in terms of the filter's frequency response H(ejω). One goal of FIR filter design is to select filter taps and filter length that minimize stopband ripple and passband ripple for the filter's frequency response. (For a digital filter, the frequency response function is the filter's transfer function H(z), evaluated on the unit circle in the complex z-plane, i.e., z=ejω) The Remez algorithm is often used in FIR filter design to solve for the filter tap coefficients that produce the desired equal ripple passband and stopband behavior.
In Adams, an efficient digital FIR lowpass filter was implemented as a cascade of a multiplierless prefilter and an amplitude equalizer. Instead of building the conventional filter as dictated by the design technique used, a less costly implementation is utilized for the prefilter (e.g., all tap coefficients set to one with the first null occurring at the beginning of the stopband). The prefilter is not an optimal filter by itself. However, the cascaded amplitude equalizer fixes the imperfections of the rough pre-filter. The resulting efficient FIR lowpass filter meets the required design constraints for the FIR filter and has both a reduction in hardware implementation costs as well as improved sensitivity. Roughly speaking, the lowered sensitivity of the filter's frequency response to perturbations of the amplitude equalizer's coefficients results from the filtering action of the prefilter.
A common component in digital circuitry for communication systems is the halfband filter. Halfband filters are often used in cooperation with up-samplers and down-samplers in multirate systems when a sampling-rate change is required. Because of the requirements of a halfband filter, the type of prefilter used in Adams cannot implement a desensitized halfband filter.
When a filter removes a band of frequencies (the frequencies in the filter's stopband) from an input signal, the usual desire is that the signal's frequencies that remain (the frequencies in the filter's passband) are not altered. Insofar as the magnitude of these passband-frequency signal-components is concerned, this means that, ideally, we desire the filter's passband gain to be exactly 1, i.e., the magnitude of the filter's transfer function should be exactly 1 throughout the filter's passband. On a decibel scale, this gain of 1 is equivalent to a value of 0 dB. Usually, this “exactly 0 dB” (perfectly flat) passband is not possible to achieve, so the filter designer is required to keep the deviations away from 0 dB, for the magnitude of the filter's transfer function, sufficiently small throughout the passband.
One form of passband deviation that is often encountered is “passband droop.” This kind of deviation from the ideal occurs near the edge of the passband, where the transfer function is beginning the process of making its transition to the stopband. FIG. 38 depicts an example of droop. Here the filter is a lowpass filter and the droop occurs at the rightmost edge of the passband. If the worst-case deviation caused by the droop d2 is sufficiently small, the deviation can be deemed tolerable and no correction may be needed. Droop often becomes problematic in systems wherein a cascade of several filters operates upon an input signal. Then, the small amounts of passband droop caused by imperfections in each of the filters' passbands, add together (i.e., they add, on a dB scale) to make a situation in which the system's total droop becomes intolerable. An additional operation upon the signal being filtered, wherein a small amount of gain is provided for frequencies near the edge of the passband, can then be desirable. Such an operation is called “droop correction” or “droop compensation” and is provided by an additional element such as a droop correction filter. An example of this foam of droop correction is given in B. P. Brandt and B. Wooley, “A low-power, area-efficient digital filter for decimation and interpolation,” IEEE J. Solid-State Circuits, vol. 29, no. 6, pp. 679-687, June 1994 [Brandt]. The Droop correction of Brandt is illustrated in FIGS. 39 and 40 discussed below.
What is therefore needed are desensitized filters providing droop correction without the use of a separate droop correction element.
The present invention will now be described with reference to the accompanying drawings. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. The drawing in which an element first appears is indicated by the leftmost digit(s) in the reference number.